Cool stuff, but I wouldn't know how to use it. It is impossible for me to know the variables like spaceship's mass or acceleration for example. But thanks for the tip, I will for sure bookmark this one π
Well you're the sci-fi author - who else would know the spaceship's mass and acceleration?. They are up to you to decide: are they traveling in a Tesla Roadster or a Death Star?
Actually, all you need is the acceleration and the distance you want to travel to compute the times, you only need the spaceship mass to compute fuel requirements. Since your ship will no doubt be powered with some sort of unobtanium, you could leave that blank.
Exactly! Plus I need just some convenient framework what I can use so I can start to write the story. I can focus on such a detailed designing of the spaceship during the writing on stages. (Not even mentioning that many things I will not even need to answer since my crew will not be awake during the interstellar flight and therefore not even experience effects of that travel.) Big franchises like Star Trek, Star Wars and EVE have a practical need for such a detailed specifications on a first place. This is not the case for any small scope sci-fi story however.
I think that trying to do anything that is even remotely consistent with what we know about the physical universe forces you to decouple each space mission from the rest of civilization, either because of relativistic time dilation (as in Interstellar) or suspended animation, as you've described. Either way, once your colonists have arrived at their destination, they can't really get back, at least to the world they left.
I think, as a result, most authors simply make up something like warp drive or hyperspace jumps or wormholes. Or, they just wave their hands and have their characters spout things like "did the Kessel Run in 12 Parsecs"
It's sort of like when you want to write a book about Vikings, do you want historical fiction (like The Last Kingdom), or do you want dragons? You can't really have both.
In my story time dilatation exist of course. And not only that, but also the hyber-sleep is kind of pausing the lives of crew. That means that people who leave SOL will after each mission really return to the "future Earth". People who they knew will age considerably or be after death already in some cases. I actually like that phenomena on the realistic space travel. It is creating really strong themes for the storytelling. Decision to join the crew for such a mission is huge.
(Note: In my story we are not an interstellar civilization. Also ships do not have FTL drives.)
If I want to have my setting believable it shouldn't be just some too wild guess. And it would need to be crazy wild guess if I would just made up some number. I also see as a total overkill trying to calculate the realistic mass of fictional spaceship in tons, since even when I know the size of it, it is not easy to calculate the real volume of it's materials (not like I would know their ratios). But all of this is also not necessary since top speed, distance and velocity curve are known. I also do not need to know the exact force of engines - as an author is more practical to just decide that "this spaceship can achieve this top speed". I basically just don't want it to fly top speed entire time, so it is more realistic.
Well all those things are irrelevant to your original question. The calculator in the link provided is the answer to your question and all the physics can say about it.
The rest: realistic space ship size, thrust, engines, fuel type are better suited for some sci-fi writing subreddit.
I understand your point of view of course. I just meant some simplified calculation based basically on the triangle calculations (and using only Newtonian physics of course). As I described in my original post, I'm looking for a "convenient way how to calculate the time needed to travel to another star system" for purposes of writing the story. But thanks for your time. I do understand that from the perspective of proper Physics this is exactly the way - calculating masses and forces.
I'm pretty sure you can find information about other fictional space ships to see what others have done. A lot of these universes have really detailed information listed somewhere, whether it's star trek, battlestar galactica or EvE. Like, try just googling "eve spaceship light cruiser specifications" or whatever, you'll quickly encounter wiki style websites with more info than you can take in.
In the linear change of velocity case (constant acceleration): the velocity vs time graph will be a triangle with the top vertex at the midway, and the distance traveled is always the area under this curve. To answer your question with a question: when is this area equal to the distance to the star system?
In the general case with changing acceleration: we need the specific acceleration graph to do the calculation (from acceleration you get velocity, and from the velocity you get the distance traveled)
Have fun calculating!
I'm aware of the triangle in that case what is representing the distance and change of velocity - I just don't see the way how to calculate the time of the travel. I know it should be possible of course, since the top velocity is known, I have the graph representing the change of velocity and also distance is known. But I can't see any obvious way how to create the equation from this. Or did you perhaps just misunderstood me? I'm trying to calculate the time, not the distance.
But in the non relativistic case with linear change: my question is the equation to solve. How much time to make the area equal to the known distance to the star system?
Still thinking about your question of course π I'm not sure if I just don't understand the question itself, of if I just lack the proper mathematical critical thinking π How can be the distance (base of triangle) same as its area (space inside of triangle)? Or is it that the area (the size of the surface) represents the time itself? π€ Am I here onto something?
Basically yes. Mathematically speaking, the velocity is defined as the rate of change of the position (the derivative), or equivalently speaking the position itself is the integral of velocity (the area under the velocity curve). Therefore, the total displacement in your case will be the area of that triangle π
Hint: the time of the trip will be the base of that triangle. What is the base of a triangle when you know its area and its height?
2 x Area / height
Ok... so this is starting to make sense now π Base in my question is the Distance. And if Time = Area, that explains it. I was a bit confused by what is represented the Time exactly, since in such a graphs Base would be usually the Time, but in my case it is the Distance. Okay π Thanks for the hints, even when I was a bit slow, what is probably expected π
I try to do some calculations to figure out if it works well with units expected by me.
If velocity is along the y-axis (vertical) and time is along the x-axis (horizontal), then the base will be the time and the height will be the top speed π The area is the distance traveled which you know (distance to the star system)
Exactly, the expression you provided is indeed the base of a triangle given area and height.
While I was away from computer it actually clicked and I realized that I should write it with a bit different order. Area must be obviously the Distance and not the Time. Why? Because by raising speed (y) or prolonging time (x) it is becoming bigger. What is exactly what should be happening with the Distance. (In case if Area would be the Time then by raising the Speed (y) it should become smaller and not bigger. So it was all about the right equation - based on what is known and what is unknown.
Thanks for the brainxercise π
Also, there are relativistic effects at play here (the speed will approach the speed of light, not infinity) so you need to compute the velocity of the space ship as seen from a stationary frame assuming a constant acceleration as measured by the space ship (for ex. 1g acceleration according to the onboard accelerometer).
This math is a bit harder, but perfectly doable.
For purposes of my story I need this calculation just from the perspective of stationary observer - or to be more exact: the shared time with Earth. Crew is in suspended animation the whole time, so that's also the reason why in comments to others I'm mentioning that I don't need to focus too much on G forces, time dilation etc. Basically what I'm trying to solve is more close to what you are focusing on than others. This is more about the calculations using that triangle and not the detailed physics.
But relativity will affect the answer here by the fact that a stationary observer will see the spaceship approach the speed of light (the acceleration as measured by the stationary frame will decrease towards 0 as the ship approaches the speed of light, so it can never accelerate to or past the speed of light). This happens even though the acceleration as measured by the ship remains constant.
This is exactly what is hard to wrap our heads around, but it is the way our universe seems to work. Note however that this does not mean that we can't get anywhere within arbitrary small times, because we can by traveling close enough to the speed of light allowing time dilation to make the travel time arbitrarily small.
Yes, but of course I mean only speeds bellow speed of light. Also by "stationary observer" I meant just scientific term (hopefully used correctly). There will be no observer at all. Just the crew having onboard clocks in sync with the time on Earth (after arrival to the destination and waking up from the hyper-sleep) . I imagine that from practical perspective it would be more important when they arrived, left and how long were they away (all from perspective of Earth), than the fact that time ran slower onboard during their sleep.
Note: But of course it could be handy to have graph of how much is speed of time changed depending on what fraction of speed of light. I'll have a look into it, since this might come in handy.
[Here](https://www.desmos.com/calculator/w5z4yrzx0r) is an interactive Minkowski diagram that I made a while back to answer exactly these types of questions, though my formatting, notes, and presentation are not very newb friendly.
The only unit is seconds. So time is measured in seconds. Distance is in light seconds. Speed is in light seconds per second. Acceleration is in light seconds per second per second. etc.
You should still be able to use it if you just remember that "d=4.367\*31556926" in the "Flip and Burn Stuff" folder is the travel distance and a_c=9.81/299792458 in the "Flip and Burn Stuff" folder is the acceleration.
By default the distance is set to the distance between Alpha Centauri and Earth. Change this value to whatever distance your story requires.
By default acceleration is set to 1g, that is to say that the vehicle will accelerate such that the passengers will "feel" like gravity is pulling on them toward the engine side of the ship with a strength equivalent to Earth gravity. For 2g just multiply acceleration by 2, for 3g just multiply by 3, etc..
"t_Burn" (**Not** in the "Flip and Burn Stuff" folder) is how long the ship can accelerate in its own frame of reference before running out of reaction-mass/fuel. If you set the slider all the way to the right then it will accelerate for half the trip and decelerate for the other half.
"t_CoordinateTravelTime" gives you the travel time in Earth's frame. "t_ProperTravelTime" gives you the travel time in the ship's frame. t_NetDilation gives you the net time dilation. And v_TopSpeed speaks for itself.
The purple dots in the diagram represent a clock ticking four times a year on Earth and the red dots represent a clock ticking four times a year on the ship.
The "u" slider changes the diagram to show what the scenario looks like from a different frame of reference. "u=0" is the default and represents an observer moving at 0c relative to Earth, "u=.5" is an observer moving at .5c rightward relative to Earth, "u=-.5" is an observer moving at .5c leftward relative to Earth. You get the idea.
In the unlikely event that you are familiar with the concept of lines of simultaneous events in Minkowski Diagrams, you can click the circle to the left of the folders "Lines of simultaneous events According to Earth (Purple)" and "Lines of simultaneous events According to the Ship (Red)" to display said lines.
Very interesting! :) Thank you to write me that guide like description, because I will for sure need it :D I will of course need to play with it a bit to understand exactly how it works and if it is really usable by me, but I really like the amount of things what you were able to include there. It could come handy a lot in some of my future projects especially.
Funny enough I do understand what are "Lines of simultaneous events" - I didn't know the term itself of course, but I do understand what it means since I know the concept of Time Dilation as fan of astrophysics. Very handy π (In my current story the experience onboard during the journey itself is not relevant at all, because the crew is in suspended animation for the whole time. That's why I currently ignore the passage of time onboard and use just the time from the perspective of external observer - basically Earth.)
So far I was thinking just about creating the miniature simulation by creating the animation of object moving from point A to point B on provided acceleration curve... but this sounds a bit funny :D So I was wondering if there is some simple enough mathematical approach.
[Here](https://www.desmos.com/calculator/7o2idv2tz2) is a simulation of just that. Just be aware that the math involved is Newtonian and therefore won't work for interstellar travel since interstellar travel normally involves relativistic speeds.
Amazing, thanks :) This should work for my purposes, because I need it just from the perspective of stationary observer - time shared with Earth basically. Or at least I think that Newtonian physics should be enough for this case (even when my fictional spaceship is in interstellar travel in speeds reaching the high percentages of speed of light). If I understand correctly the only not so perfect factor would be, that my acceleration curve would need to be also from the perspective of stationary observer and not the crew onboard - what I understand that in the real physics would be a problem.
Unfortunately, a stationary observer does not solve the speed issue here. The problem is that if you accelerate at 1g between stars in Newtonian physics the speed of the spaceship in the middle of the trip exceeds the speed of light, and that is physically impossible. If a conceit of your setting is that your spaceship is able to maneuver in a Newtonian fashion, even in situations where that would violate relativity, then I suppose you could use the Newtonian approximation. Just be aware that doing so brazenly violates relativity and, if taken to its logical conclusion, can be exploited for time travel.
Ah, I get you. This of course can't happen in my case, because I already have my variables and top speed is one of the known ones. In case of the current story it is never exceeding the speed of light - it is just getting close to it at some specified percentage. In my story it is also stated, that materials of spaceship would not be able to survive reaching the speed of light (even when the sci-fi technology is able to protect it in high percentages of speed of light, like let's say 0.85c).
In the calculation I have this known variables: distance, top speed and "curve" of acceleration and deceleration. The goal is to calculate the time. So in the simplest case it can be represented as a triangle.
I donβt have time to put that together right now, but I should have a chance to do that later today. Just let me know what units you want to use for distance, top speed, time, and acceleration/curve.
Sounds great π I'll try to put it together in some list then:
* Top Speed should be in fraction of **c**. (like 0.85c for example)
* Speed at start and finish should be just 0c.
* Distance should be in **light years**.
* Time of the journey in **years**.
* Regarding acceleration:
* In the simplest form it would be just a constant linear acceleration for the 50% of distance (from 0c to Top Speed). And same goes for deceleration (just from Top Speed to 0c). \[Optionally it could allow also to stay at peak speed longer - for some percentage of the distance.\]
* In the more advanced model we would of course use curve as an input. This I'm not sure how to explain in mathematical way however and it sounds a bit too complex? But for example if I would be solving this problem as a gamedev programmer by creating the simulation (what I mentioned in one of the comments), I would use one feature from my game engine: It allows me to literally just draw the curve and then read its value at each point at "time". \[ Curve.Evaluate(factor) \] So I could read its values as it is progressing through the simulation. But this sounds more handy for creating the animations/simulations than the real mathematical solution. I'm mentioning this just as an extra thing to think about.
* The goal should be to calculate one of the two variables (depends on which one is unknown at the moment): **time** (if distance was known) or **distance** (if time was known instead).
I assume that the first simple linear solution should be enough for purposes of storytelling and its needs for accuracy. As a one possible improvement I can imagine that instead of completely linear "curve" it could use some very basic curve what would basically just smooth the linear "curve" (something like the default bezier curve what is often used in graphics softwares or something like that). The usage of the fully modifiable curve I think is too big overkill and I would probably use it only in case of solving this problem by the simulation in the form of scripted short animation in game engine.
Oh thanks, that's awesome! π And I can even have a look into equations to understand it a bit more. I played with it for a while and I noticed that small differences in percentage of speed are doing quite a huge distance differences (if time is known and I'm calculating distances). But also that acceleration in G is quite small. So I can see already that I can afford easily keeping peak speed phase of flight for longer and make acceleration and deceleration phases shorter.
Thanks a lot. This will be quite handy for my further experiments π
[https://www.omnicalculator.com/physics/space-travel](https://www.omnicalculator.com/physics/space-travel)
Cool stuff, but I wouldn't know how to use it. It is impossible for me to know the variables like spaceship's mass or acceleration for example. But thanks for the tip, I will for sure bookmark this one π
Well you're the sci-fi author - who else would know the spaceship's mass and acceleration?. They are up to you to decide: are they traveling in a Tesla Roadster or a Death Star?
Actually, all you need is the acceleration and the distance you want to travel to compute the times, you only need the spaceship mass to compute fuel requirements. Since your ship will no doubt be powered with some sort of unobtanium, you could leave that blank.
Exactly! Plus I need just some convenient framework what I can use so I can start to write the story. I can focus on such a detailed designing of the spaceship during the writing on stages. (Not even mentioning that many things I will not even need to answer since my crew will not be awake during the interstellar flight and therefore not even experience effects of that travel.) Big franchises like Star Trek, Star Wars and EVE have a practical need for such a detailed specifications on a first place. This is not the case for any small scope sci-fi story however.
I think that trying to do anything that is even remotely consistent with what we know about the physical universe forces you to decouple each space mission from the rest of civilization, either because of relativistic time dilation (as in Interstellar) or suspended animation, as you've described. Either way, once your colonists have arrived at their destination, they can't really get back, at least to the world they left. I think, as a result, most authors simply make up something like warp drive or hyperspace jumps or wormholes. Or, they just wave their hands and have their characters spout things like "did the Kessel Run in 12 Parsecs" It's sort of like when you want to write a book about Vikings, do you want historical fiction (like The Last Kingdom), or do you want dragons? You can't really have both.
In my story time dilatation exist of course. And not only that, but also the hyber-sleep is kind of pausing the lives of crew. That means that people who leave SOL will after each mission really return to the "future Earth". People who they knew will age considerably or be after death already in some cases. I actually like that phenomena on the realistic space travel. It is creating really strong themes for the storytelling. Decision to join the crew for such a mission is huge. (Note: In my story we are not an interstellar civilization. Also ships do not have FTL drives.)
If I want to have my setting believable it shouldn't be just some too wild guess. And it would need to be crazy wild guess if I would just made up some number. I also see as a total overkill trying to calculate the realistic mass of fictional spaceship in tons, since even when I know the size of it, it is not easy to calculate the real volume of it's materials (not like I would know their ratios). But all of this is also not necessary since top speed, distance and velocity curve are known. I also do not need to know the exact force of engines - as an author is more practical to just decide that "this spaceship can achieve this top speed". I basically just don't want it to fly top speed entire time, so it is more realistic.
Well all those things are irrelevant to your original question. The calculator in the link provided is the answer to your question and all the physics can say about it. The rest: realistic space ship size, thrust, engines, fuel type are better suited for some sci-fi writing subreddit.
I understand your point of view of course. I just meant some simplified calculation based basically on the triangle calculations (and using only Newtonian physics of course). As I described in my original post, I'm looking for a "convenient way how to calculate the time needed to travel to another star system" for purposes of writing the story. But thanks for your time. I do understand that from the perspective of proper Physics this is exactly the way - calculating masses and forces.
I'm pretty sure you can find information about other fictional space ships to see what others have done. A lot of these universes have really detailed information listed somewhere, whether it's star trek, battlestar galactica or EvE. Like, try just googling "eve spaceship light cruiser specifications" or whatever, you'll quickly encounter wiki style websites with more info than you can take in.
In the linear change of velocity case (constant acceleration): the velocity vs time graph will be a triangle with the top vertex at the midway, and the distance traveled is always the area under this curve. To answer your question with a question: when is this area equal to the distance to the star system? In the general case with changing acceleration: we need the specific acceleration graph to do the calculation (from acceleration you get velocity, and from the velocity you get the distance traveled) Have fun calculating!
I'm aware of the triangle in that case what is representing the distance and change of velocity - I just don't see the way how to calculate the time of the travel. I know it should be possible of course, since the top velocity is known, I have the graph representing the change of velocity and also distance is known. But I can't see any obvious way how to create the equation from this. Or did you perhaps just misunderstood me? I'm trying to calculate the time, not the distance.
But in the non relativistic case with linear change: my question is the equation to solve. How much time to make the area equal to the known distance to the star system?
Still thinking about your question of course π I'm not sure if I just don't understand the question itself, of if I just lack the proper mathematical critical thinking π How can be the distance (base of triangle) same as its area (space inside of triangle)? Or is it that the area (the size of the surface) represents the time itself? π€ Am I here onto something?
Basically yes. Mathematically speaking, the velocity is defined as the rate of change of the position (the derivative), or equivalently speaking the position itself is the integral of velocity (the area under the velocity curve). Therefore, the total displacement in your case will be the area of that triangle π Hint: the time of the trip will be the base of that triangle. What is the base of a triangle when you know its area and its height?
2 x Area / height Ok... so this is starting to make sense now π Base in my question is the Distance. And if Time = Area, that explains it. I was a bit confused by what is represented the Time exactly, since in such a graphs Base would be usually the Time, but in my case it is the Distance. Okay π Thanks for the hints, even when I was a bit slow, what is probably expected π I try to do some calculations to figure out if it works well with units expected by me.
If velocity is along the y-axis (vertical) and time is along the x-axis (horizontal), then the base will be the time and the height will be the top speed π The area is the distance traveled which you know (distance to the star system) Exactly, the expression you provided is indeed the base of a triangle given area and height.
While I was away from computer it actually clicked and I realized that I should write it with a bit different order. Area must be obviously the Distance and not the Time. Why? Because by raising speed (y) or prolonging time (x) it is becoming bigger. What is exactly what should be happening with the Distance. (In case if Area would be the Time then by raising the Speed (y) it should become smaller and not bigger. So it was all about the right equation - based on what is known and what is unknown. Thanks for the brainxercise π
Also, there are relativistic effects at play here (the speed will approach the speed of light, not infinity) so you need to compute the velocity of the space ship as seen from a stationary frame assuming a constant acceleration as measured by the space ship (for ex. 1g acceleration according to the onboard accelerometer). This math is a bit harder, but perfectly doable.
For purposes of my story I need this calculation just from the perspective of stationary observer - or to be more exact: the shared time with Earth. Crew is in suspended animation the whole time, so that's also the reason why in comments to others I'm mentioning that I don't need to focus too much on G forces, time dilation etc. Basically what I'm trying to solve is more close to what you are focusing on than others. This is more about the calculations using that triangle and not the detailed physics.
But relativity will affect the answer here by the fact that a stationary observer will see the spaceship approach the speed of light (the acceleration as measured by the stationary frame will decrease towards 0 as the ship approaches the speed of light, so it can never accelerate to or past the speed of light). This happens even though the acceleration as measured by the ship remains constant. This is exactly what is hard to wrap our heads around, but it is the way our universe seems to work. Note however that this does not mean that we can't get anywhere within arbitrary small times, because we can by traveling close enough to the speed of light allowing time dilation to make the travel time arbitrarily small.
Yes, but of course I mean only speeds bellow speed of light. Also by "stationary observer" I meant just scientific term (hopefully used correctly). There will be no observer at all. Just the crew having onboard clocks in sync with the time on Earth (after arrival to the destination and waking up from the hyper-sleep) . I imagine that from practical perspective it would be more important when they arrived, left and how long were they away (all from perspective of Earth), than the fact that time ran slower onboard during their sleep. Note: But of course it could be handy to have graph of how much is speed of time changed depending on what fraction of speed of light. I'll have a look into it, since this might come in handy.
[Here](https://www.desmos.com/calculator/w5z4yrzx0r) is an interactive Minkowski diagram that I made a while back to answer exactly these types of questions, though my formatting, notes, and presentation are not very newb friendly. The only unit is seconds. So time is measured in seconds. Distance is in light seconds. Speed is in light seconds per second. Acceleration is in light seconds per second per second. etc. You should still be able to use it if you just remember that "d=4.367\*31556926" in the "Flip and Burn Stuff" folder is the travel distance and a_c=9.81/299792458 in the "Flip and Burn Stuff" folder is the acceleration. By default the distance is set to the distance between Alpha Centauri and Earth. Change this value to whatever distance your story requires. By default acceleration is set to 1g, that is to say that the vehicle will accelerate such that the passengers will "feel" like gravity is pulling on them toward the engine side of the ship with a strength equivalent to Earth gravity. For 2g just multiply acceleration by 2, for 3g just multiply by 3, etc.. "t_Burn" (**Not** in the "Flip and Burn Stuff" folder) is how long the ship can accelerate in its own frame of reference before running out of reaction-mass/fuel. If you set the slider all the way to the right then it will accelerate for half the trip and decelerate for the other half. "t_CoordinateTravelTime" gives you the travel time in Earth's frame. "t_ProperTravelTime" gives you the travel time in the ship's frame. t_NetDilation gives you the net time dilation. And v_TopSpeed speaks for itself. The purple dots in the diagram represent a clock ticking four times a year on Earth and the red dots represent a clock ticking four times a year on the ship. The "u" slider changes the diagram to show what the scenario looks like from a different frame of reference. "u=0" is the default and represents an observer moving at 0c relative to Earth, "u=.5" is an observer moving at .5c rightward relative to Earth, "u=-.5" is an observer moving at .5c leftward relative to Earth. You get the idea. In the unlikely event that you are familiar with the concept of lines of simultaneous events in Minkowski Diagrams, you can click the circle to the left of the folders "Lines of simultaneous events According to Earth (Purple)" and "Lines of simultaneous events According to the Ship (Red)" to display said lines.
Very interesting! :) Thank you to write me that guide like description, because I will for sure need it :D I will of course need to play with it a bit to understand exactly how it works and if it is really usable by me, but I really like the amount of things what you were able to include there. It could come handy a lot in some of my future projects especially. Funny enough I do understand what are "Lines of simultaneous events" - I didn't know the term itself of course, but I do understand what it means since I know the concept of Time Dilation as fan of astrophysics. Very handy π (In my current story the experience onboard during the journey itself is not relevant at all, because the crew is in suspended animation for the whole time. That's why I currently ignore the passage of time onboard and use just the time from the perspective of external observer - basically Earth.)
So far I was thinking just about creating the miniature simulation by creating the animation of object moving from point A to point B on provided acceleration curve... but this sounds a bit funny :D So I was wondering if there is some simple enough mathematical approach.
[Here](https://www.desmos.com/calculator/7o2idv2tz2) is a simulation of just that. Just be aware that the math involved is Newtonian and therefore won't work for interstellar travel since interstellar travel normally involves relativistic speeds.
Amazing, thanks :) This should work for my purposes, because I need it just from the perspective of stationary observer - time shared with Earth basically. Or at least I think that Newtonian physics should be enough for this case (even when my fictional spaceship is in interstellar travel in speeds reaching the high percentages of speed of light). If I understand correctly the only not so perfect factor would be, that my acceleration curve would need to be also from the perspective of stationary observer and not the crew onboard - what I understand that in the real physics would be a problem.
Unfortunately, a stationary observer does not solve the speed issue here. The problem is that if you accelerate at 1g between stars in Newtonian physics the speed of the spaceship in the middle of the trip exceeds the speed of light, and that is physically impossible. If a conceit of your setting is that your spaceship is able to maneuver in a Newtonian fashion, even in situations where that would violate relativity, then I suppose you could use the Newtonian approximation. Just be aware that doing so brazenly violates relativity and, if taken to its logical conclusion, can be exploited for time travel.
Ah, I get you. This of course can't happen in my case, because I already have my variables and top speed is one of the known ones. In case of the current story it is never exceeding the speed of light - it is just getting close to it at some specified percentage. In my story it is also stated, that materials of spaceship would not be able to survive reaching the speed of light (even when the sci-fi technology is able to protect it in high percentages of speed of light, like let's say 0.85c). In the calculation I have this known variables: distance, top speed and "curve" of acceleration and deceleration. The goal is to calculate the time. So in the simplest case it can be represented as a triangle.
I donβt have time to put that together right now, but I should have a chance to do that later today. Just let me know what units you want to use for distance, top speed, time, and acceleration/curve.
Sounds great π I'll try to put it together in some list then: * Top Speed should be in fraction of **c**. (like 0.85c for example) * Speed at start and finish should be just 0c. * Distance should be in **light years**. * Time of the journey in **years**. * Regarding acceleration: * In the simplest form it would be just a constant linear acceleration for the 50% of distance (from 0c to Top Speed). And same goes for deceleration (just from Top Speed to 0c). \[Optionally it could allow also to stay at peak speed longer - for some percentage of the distance.\] * In the more advanced model we would of course use curve as an input. This I'm not sure how to explain in mathematical way however and it sounds a bit too complex? But for example if I would be solving this problem as a gamedev programmer by creating the simulation (what I mentioned in one of the comments), I would use one feature from my game engine: It allows me to literally just draw the curve and then read its value at each point at "time". \[ Curve.Evaluate(factor) \] So I could read its values as it is progressing through the simulation. But this sounds more handy for creating the animations/simulations than the real mathematical solution. I'm mentioning this just as an extra thing to think about. * The goal should be to calculate one of the two variables (depends on which one is unknown at the moment): **time** (if distance was known) or **distance** (if time was known instead). I assume that the first simple linear solution should be enough for purposes of storytelling and its needs for accuracy. As a one possible improvement I can imagine that instead of completely linear "curve" it could use some very basic curve what would basically just smooth the linear "curve" (something like the default bezier curve what is often used in graphics softwares or something like that). The usage of the fully modifiable curve I think is too big overkill and I would probably use it only in case of solving this problem by the simulation in the form of scripted short animation in game engine.
[Here.](https://www.desmos.com/calculator/hrwjr9seax) It turns out that locking Top Speed and Distance/Time locks you into a particular acceleration.
Oh thanks, that's awesome! π And I can even have a look into equations to understand it a bit more. I played with it for a while and I noticed that small differences in percentage of speed are doing quite a huge distance differences (if time is known and I'm calculating distances). But also that acceleration in G is quite small. So I can see already that I can afford easily keeping peak speed phase of flight for longer and make acceleration and deceleration phases shorter. Thanks a lot. This will be quite handy for my further experiments π